245 research outputs found
Value Iteration Using Universal Graphs and the Complexity of Mean Payoff Games
We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in NP ? coNP and not known to be in P. In a breakthrough result Calude, Jain, Khoussainov, Li, and Stephan constructed in 2017 a quasipolynomial time algorithm for solving parity games, which was quickly followed by a few other algorithms with the same complexity. Our objective is to investigate how these techniques can be extended to mean payoff games.
The starting point is the combinatorial notion of universal trees: all quasipolynomial time algorithms for parity games have been shown to exploit universal trees. Universal graphs extend universal trees to arbitrary (positionally determined) objectives. We show that they yield a family of value iteration algorithms for solving mean payoff games which includes the value iteration algorithm due to Brim, Chaloupka, Doyen, Gentilini, and Raskin.
The contribution of this paper is to prove tight bounds on the complexity of algorithms for mean payoff games using universal graphs. We consider two parameters: the largest weight N in absolute value and the number k of weights. The dependence in N in the existing value iteration algorithm is linear, we show that this can be improved to N^{1 - 1/n} and obtain a matching lower bound. However, we show that we cannot break the linear dependence in the exponent in the number k of weights implying that universal graphs do not yield a quasipolynomial time algorithm for solving mean payoff games
Expectations or Guarantees? I Want It All! A crossroad between games and MDPs
When reasoning about the strategic capabilities of an agent, it is important
to consider the nature of its adversaries. In the particular context of
controller synthesis for quantitative specifications, the usual problem is to
devise a strategy for a reactive system which yields some desired performance,
taking into account the possible impact of the environment of the system. There
are at least two ways to look at this environment. In the classical analysis of
two-player quantitative games, the environment is purely antagonistic and the
problem is to provide strict performance guarantees. In Markov decision
processes, the environment is seen as purely stochastic: the aim is then to
optimize the expected payoff, with no guarantee on individual outcomes.
In this expository work, we report on recent results introducing the beyond
worst-case synthesis problem, which is to construct strategies that guarantee
some quantitative requirement in the worst-case while providing an higher
expected value against a particular stochastic model of the environment given
as input. This problem is relevant to produce system controllers that provide
nice expected performance in the everyday situation while ensuring a strict
(but relaxed) performance threshold even in the event of very bad (while
unlikely) circumstances. It has been studied for both the mean-payoff and the
shortest path quantitative measures.Comment: In Proceedings SR 2014, arXiv:1404.041
New Algorithms for Solving Tropical Linear Systems
The problem of solving tropical linear systems, a natural problem of tropical
mathematics, has already proven to be very interesting from the algorithmic
point of view: it is known to be in but no polynomial time
algorithm is known, although counterexamples for existing pseudopolynomial
algorithms are (and have to be) very complex.
In this work, we continue the study of algorithms for solving tropical linear
systems. First, we present a new reformulation of Grigoriev's algorithm that
brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us
formulate a whole family of new algorithms, and we present algorithms from this
family for which no known superpolynomial counterexamples work. Second, we
present a family of algorithms for solving overdetermined tropical systems. We
show that for weakly overdetermined systems, there are polynomial algorithms in
this family. We also present a concrete algorithm from this family that can
solve a tropical linear system defined by an matrix with maximal
element in time , and this time matches the complexity of the best of
previously known algorithms for feasibility testing.Comment: 17 page
Symmetric Strategy Improvement
Symmetry is inherent in the definition of most of the two-player zero-sum
games, including parity, mean-payoff, and discounted-payoff games. It is
therefore quite surprising that no symmetric analysis techniques for these
games exist. We develop a novel symmetric strategy improvement algorithm where,
in each iteration, the strategies of both players are improved simultaneously.
We show that symmetric strategy improvement defies Friedmann's traps, which
shook the belief in the potential of classic strategy improvement to be
polynomial
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